Mixed Integer Models for the Optimisation of Gas Networks in the Stationary Case

Language:

English

Abstract:

Through out the world the natural gas resources will be one of the most important sources of energy in the future. The development of optimised possibilities for the distribution of gas through a network of pipelines will be very important for an effective operation of a gas transmission network. The aim of this thesis is to formulate this problem as a suitable mathematical mixed integer problem and to find advanced solutions, using techniques of mixed integer programming. The main problem of the so called Transient Technical Optimisation (TTO) is to minimise the total supply costs of a gas transmission company that has to satisfy demands of different kinds. A gas network basically consists of a number of compressors and valves that are connected via pipes. The gas transmission companies dispatchers decide how to run the compressors and how to switch the valves cost-efficiently such that all demands of all customers are satisfied. The cost function mainly consists of the supply costs of driving the compressors. Note that the compressors consum a fraction of the gas transported through the pipelines. The costs imposed by consumed gas should be minimised. The gas transmission network has to satisfy several demands that are described by a minimal or maximal pressure requirement at a certain node or in a pipe. Also the consumers want to get gas of a certain volume and quality. Furthermore some physical constraints, like Kirchhoff's laws have to be modelled. There are also some combinatorial constraints, e.g. the different possibilities of switching the valves or compressor configurations. Note, that some of the constraints are nonlinear, like the pressure loss in a pipeline or the fuel-gas consumption of the compressors. In order to formulate TTO as a mixed integer program we approximate the nonlinear constraints by piecewise linear functions. Considering the experiences of other projects where mixed integer programs have been used, e.g. VLSI-Design or Telecommunications, we know that the problem can be solved by examination of the underlying polyhedra of such complex and high-dimensional mixed integer programs. We know from earlier test evaluations of smaller problems that it is not possible to solve real gas transmission problems with state-of-the-art general mixed integer programming solvers. One programming approach is the search of better valid (or even facet-defining) inequalities of the polyhedra for the use in a Branch-and-Cut Algorithm. We have developed a new class of valid inequalities that have been integrated in a general MIP solver algorithm. Summarising the results it was possible to develop a polynomial separation algorithm for a special class of polyhedra. The use of these cuts reduces the calculation time by a significant factor. A suitable branch-and-bound algorithm is also added. The cuts and the branching algorithms have been tested on several test-models of real gas-networks.